Decision, Risk & Operations Working Papers Series

Dynamic pricing when customers strategically time their purchase M. Bansal and C. Maglaras July 2008 DRO-2008-03

Dynamic pricing when customers strategically

time their purchase

Matulya Bansal Costis Maglaras∗

July 21, 2008

We study the dynamic pricing problem of a monopolist firm in presence of strategic cus-

tomers that differ in their valuations and risk-preferences. We show that this problem can be

formulated as a static mechanism design problem, which is more amenable to analysis. We

highlight several structural properties of the optimal solution, and solve the problem for sev-

eral special cases. Focusing on settings with low risk-aversion, we show through an asymptotic

analysis that the “two-price point” strategy is near-optimal, offering partial validation for its

wide use in practice, but also highlighting when it is indeed suitable to adopt it.

1 Introduction

The wide adoption of promotional and markdown pricing by major retailers has “trained” many

consumers to anticipate these events and accordingly time their purchases. Given this observa-

tion, a natural question that arises is the following: How should the retailer price and allocate

its inventory over time in presence of customers that strategize their purchasing decisions? Most

pricing and revenue management models and associated commercial systems do not explicitly

incorporate this level of consumer behavior. This paper is part of a growing literature that tries

to model this effect and study its impact on the firm’s controls and profitability.

In more detail, we consider a revenue-maximizing monopolist firm, the seller, that sells a

homogeneous good over a time horizon to a market of heterogeneous strategic customers that

differ in their valuations and risk-preferences. We allow for a discrete (but arbitrary) customer

valuation distribution. The firm seeks to discriminate customers by selling the product at

different points in time at different prices and fill-rates. This creates rationing risk, i.e., the

risk of not being able to procure the product because its availability is limited, and introduces

an incentive for customers with higher valuations, or that are more risk-averse, to pay more∗Both authors are with Graduate School of Business, Columbia University, 3022 Broadway, New York, NY

10027. Email: mbansal11@gsb.columbia.edu and c.maglaras@gsb.columbia.edu

1

for the product offered during periods with higher availability. Customers observe or anticipate

correctly the price and associated product availability at different points in time and decide

when, if at all, to attempt to purchase the product in a way that maximizes their expected risk-

adjusted, net utility from their purchase. The seller knows the market characteristics, i.e., the

number of customer types, the market size of each type, which is assumed to be deterministic,

its value and risk preferences, but cannot discriminate across types. The seller’s problem is

to choose its dynamic pricing and product availability strategies to maximize its profitability

taking into account the strategic customer choice behavior.

The key contributions of this paper are two. First, it highlights a connection between the

seller’s dynamic pricing problem and a static product design problem, where the firm selects an

optimal menu of (price, rationing probability) combinations to optimally segment the market.

This connection allows one to use standard machinery from mechanism design to study the

structural properties of the seller’s policy; it also provides a novel way of addressing the dynamic

pricing problem that can be extended in several ways. The second contribution of the paper

is that it shows when is it optimal or near-optimal to adopt a two product heuristic, which if

often used in the literature to simplify analysis and due to its practical appeal. To this point,

it is optimal to offer a menu with one or two products when all customers are risk-neutral,

and it is asymptotically optimal to follow a similar strategy even when customers have low risk

aversion. The latter case asymptotically reduces to the hierarchical solution of two LPs:

1. The first LP solves the seller’s problem treating all consumer types as being risk-neutral.

2. The second LP solves for price and rationing risk perturbations around the risk-neutral

solution, taking into account the risk-aversion of the various customer types.

The above decomposition is justified asymptotically as the risk-aversion coefficients of all market

participants approach one (i.e., the risk-neutral case). The optimal perturbation around the

optimal risk neutral product menu retains the two product structure, but appropriately adjusts

the price and rationing probabilities according to the “linearized” risk preferences of the various

customer segments. In that sense the asymptotic optimality of the two-product solution is

fairly robust. This lends credibility to a practical heuristic that would focus on identifying the

optimal two product offering, which as we show can be solved very efficiently even without this

asymptotic decomposition. Numerical results are used to benchmark this heuristic against the

brute-force computational solution.

Our analysis partially extends prior work that has made restrictive assumptions with respect

to customer heterogeneity, e.g., assuming two discrete types of customers, uniformly distributed

valuations, no price control, and/or focus on two product variants (i.e., product offered at only

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two price points). The extension to multiple types and multiple products comes at the expense

of the assumption that customers that are rationed out do not renter the market in future

times. This assumption is not needed if the market has two types, or if the seller selects to offer

just two products. In the general case, it can be justified if customers incur important fixed,

transportation or other overhead and search costs in making subsequent store visits, or if they

prefer outside opportunities upon being rationed out.

The remainder of this paper is organized as follows: this section concludes with a brief

literature review. In §2, we pose the firm’s dynamic pricing problem, and develop its reformu-

lation as a product design problem. §3 characterizes the structure of the optimal solution, and

§4 solves the two-product problem, the problem with risk-neutral customers, §5 addresses the

problem with low risk-aversion, and §6 presents some numerical results.

Literature review: The economic literature that accounts for the effect of strategic consumer

behavior in the context of revenue optimization for a monopolist dates back to Coase’s [4]

treatment of the durable good problem; see also Bulow [1]. We refer the reader to the recent

papers by Liu and van Ryzin [8] and Shen and Su [13] for thorough reviews of the extensive

economic literature in this area.

Our paper is more closely related to two sets of papers that focus on revenue optimization

problems of perishable products with strategic consumers, which is in part motivated by the

impact of this type of behavior in retailing coupled with the proliferation of revenue management

(price markdown) systems in this industry. This literature is also reviewed in [8] and [13].

The first studies questions of dynamic pricing optimization in a variety of settings that are

differentiated by the assumptions on customer heterogeneity. Liu and van Ryzin [8] studied

the problem of offering two product variants at predetermined prices to a market of risk-

averse consumers with uniformly distributed valuations. Other related papers include Su [14]

looks at a problem with high-valuation and low-valuation customers (i.e with a two-point mass

valuation distribution) that are either strategic or myopic (purchase at their time of arrival, if

at all). Cachon and Swinney [2] study the problem of offering two product variants to a market

of myopic, strategic and bargain-hunting customers (that only purchase if the price is low).

Caldentey and Vulcano [3] also considered an essentially two period problem where consumers

could choose whether to purchase in an auction or at a list price at a future time, and explicitly

modeled the possibility of purchasing in the open market at a list price if their bid was not

accepted in the auction. Zhang and Cooper [15] consider the two-period, two-product problem

with strategic and myopic customers under a linear and multiplicative demand model. While

we do not model myopic or bargain-hunting behavior as in Su [14], Cachon and Swinney [2] and

Zhang and Cooper [15], we allow customer valuations to have any arbitrary discrete distribution,

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and model risk-aversion. Unlike some of the abovementioned papers, we assume that customers

that are rationed out do not re-enter the market in future times, but on the hand extend our

treatment to problems with multiple types of customers, multiple product variants, and, in

general, time horizons with more than two periods. Specifically, this restrictive assumption is

not needed if there are only two customer types, or if the number of offered products is two.

We extend the analysis in Liu and van Ryzin [8] to arbitrary discrete valuation distribution,

allowing both prices and fill-rates to be optimization variables, and for consumers to also differ

in terms of their risk preferences. We show that several of their insights carry through in this

more general setting (such as the optimality of offering at most two products to a market of risk

neutral customers). All of the above papers studied models with two periods, or equivalently

where the seller offered the good in two (price, availability) variants. This paper lends credibility

to this modeling choice by proving that this restriction is near-optimal in problems with low

risk aversion. Specifically, the optimal product menu in settings with low risk aversion retains

the two-product structure of the risk-neutral solution but perturbs it appropriately to take

into account the risk preferences of the market. The solution of the LP that characterizes the

optimal perturbation of the two product menu is quite different from the risk-neutral product

design problem (which can also be reduced to an LP), it takes into account the “linearized”

risk preferences of the market, and the set of feasible perturbations includes -or better yet is

dominated- by price and rationing vectors that would lead to a menu with more products than

just two. The fact that the optimal perturbation retains the two product structure of the risk-

neutral solution supports the robustness of that policy. This insight is also robust to the form

in which the risk preferences enter the utility calculation. Finally, this result relies on a form

of asymptotic analysis that is novel in this literature.

The second set of papers that is related to our work is one that uses ideas from mechanism

design in the study of this type of a problem. In this paper, the connection with mechanism

design allows us to unify and extend several previously established results under a common and

intuitive framework. While not considered in this paper, the mechanism design formulation

could also be extended to include myopic customer behavior, discounted customer utility, or

heterogenous outside opportunity. Our use of the direct revelation principle (Myerson [12])

towards solving the product design reformulation of the dynamic pricing problem is very similar

to the approach adopted in Harris and Raviv [7] and Moorthy [10]. Our notion of fill-rate

corresponds to their notion of quality. In both Harris and Raviv [7] and Moorthy [10], offered

qualities affect cost and thus the revenues. However, customer utility is separable in product

price and quality. In our case fill-rates and prices enter the objective multiplicatively, leading

to a bilinear objective. This, in addition to the risk-averse behavior of customers, makes

our problem different and more complicated. Another paper that uses the direct revealation

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principle to characterize the optimal mechanism for the seller is Gallien [6], where the revenue

maximization problem of a monopolist firm operating in market of risk-neutral, time-sensitive

customers with linear utility, and in which the customer arrival process is given by a renewal

process is solved using dynamic programming. A recent preprint by Akan. et.al. [9] adopts a

mechanism design for a market with a continuum of types that know their type at the beginning

of the sales horizon but learn their true valuation at some future point that depends on their

type. Customers are strategic and the goal is to design the optimal mechanism to maximize the

sellers revenue, which ends up involving an appropriately designed set of options. [6] and [9]

study models that differ than ours in many respects, but share in common a problem formulation

that exploits a mechanism design structure.

Finally, we note that strategic rationing as a way to differentiate customers is also discussed

in Dana [5], where, however, the primary motivation for rationing is demand uncertainty.

2 Dynamic pricing with strategic customers

In this section, we formulate the firm’s dynamic pricing problem, and show how it can be refor-

mulated as a static mechanism design problem, thereby making it more amenable to analysis.

2.1 Problem formulation

Seller: A monopolist firm seeks to sell a homogeneous good to a market of strategic customers

that differ in their valuations and risk-aversion. In order to segment the market and maximize

revenue, the firm sells the product over a period of time by varying product price and the

associated fill-rate. Time is discrete, and the selling horizon is divided into T periods, indexed

by t = 1, ..., T . The capacity, denoted by C, can be endogenous (an optimization variable) or

exogenously given (fixed). The capacity cost is linear, and there is no inventory carrying cost.

We denote by pt and rt the price and the fill-rate associated with the product offered by this

monopolist in the tth period. We also refer to it as the tth product.

The firm’s policy (p, r) is assumed to be known to the customers, either because it is

announced to the market, or because customers have estimated it through repeated interactions

with the firm. The firm sells to the market in every season, and as a consequence, the firm’s

strategy (p, r) over any season should be credible in the sense that the firm commits to it at

the start of the selling horizon and cannot deviate from it at any point after that even if that

would be optimal from that instant onwards; e.g., the firm cannot announce that the low price

product variant will be offered with a significant rationing risk (i.e., at a low fill-rate), and then

once the high valuation customers buy the high price variant, decide to offer the lower priced

5

variant at full availability so as to capture more revenue.

Customers: We allow the customer valuation distribution to be arbitrary but discrete. We

assume that there are N distinct customer valuations, v1, v2, ..., vN . Without loss of generality,

we assume that T ≥ N , else the firm can further divide the time-horizon. The discrete valuations

could be obtained as a result of some clustering analysis or by dividing the support of valuation

distribution uniformly. Corresponding to each valuation vi, there is an associated number πi,

denoting the size of customer segment with this valuation. The pair (v, π) defines an arbitrary

discrete valuation distribution in a market with N types. We assume that the number of

customers πi with valuation vi is deterministic and known to the firm. This can be viewed as a

large market approximation of a stochastic model where the firm knows the stochastic demand

primitives. 1

For notational convenience, we will assume v1 > v2 > ... > vN > 0 and refer to the customer

segment that has valuation vi as “type i”. Type i customers, apart from their valuation, are

also characterized by a risk-aversion parameter, γi, assumed to be rational, and are endowed

with the power-utility function. Specifically, the net expected utility for a type i customer from

product t is given by (vi − pt)γirt. We also assume that higher valuation types are at least as

risk-averse as the low valuation types, i.e., 0 < γ1 ≤ γ2 ≤ ... ≤ γN ≤ 1. The setting where all

customers have the same degree of risk-aversion is a special case. Customers seek to purchase

a product as long as their net expected utility is non-negative. If vi < pt, then we define the

resulting utility to be 0−. Customers choose the variant that maximizes their expected net

utility according to:

χ(i, p, r) =

arg max1≤t≤T (vi − pt)γi rt, if(vi − pt)γi rt ≥ 0 for some t,

0, otherwise.(1)

If two or more products result in the same utility, we assume that customers prefer the one with

the highest fill-rate amongst these products. In the following, we will often abbreviate χ(i, p, r)

to χ(i). Note that given the discrete type space and the assumption that customers of each type

are homogeneous, all customers of each particular type will make the same choice. We assume

that each customer makes the decision to buy one of the offered products only once and buys

only one unit of product. In particular, if a customer decides to enter the system in a particular

period and does not obtain a unit of the product (because of being rationed out), then the

customer leaves and does not contend to buy any other product offered by this firm. This is

possible, for example, if customers incur transportation or overhead costs in making subsequent

attempts to purchase, or if they prefer to purchase a substitute upon being rationed out. In a1A more complex model could allow for the πi’s to be uncertain, perhaps driven by some aggregate source

of uncertainty that would translate the distribution towards lower or higher valuations, and where the firm, andperhaps the consumers, would learn over time. A two-period version of this problem was analyzed in [2].

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more general formulation, this customer could be expected to attempt to buy the product in

a later period. However, we do not model this flexibility. We note that this restriction is not

needed if the number of types is two, or if the seller restricts attention to offering at most two

products. We also assume that there is no strategic interaction amongst the customers.

Dynamic pricing problem formulation: Under the above assumptions, the revenue maxi-

mization problem faced by the monopolist is given by:

maxp,r

ΣTt=1 (ΣN

i=11{χ(i)=t}πi) rt pt (2)

s.t. ΣTt=1 (ΣN

i=11{χ(i)=t}πi) rt ≤ C, (3)

0 ≤ rt ≤ 1, 0 ≤ pt, t = 1, 2, ..., T. (4)

The objective is the sum of the revenues from all product variants: (ΣNi=11{χ(i)=t}πi) is the

number of customers that wish to purchase in period t, rt is the fraction of customers that are

served, and pt is the price per unit sold. For each time period t, the price, fill-rate combination

(pt, rt) can be interpreted as a “product” offered by the firm. The T products are sequenced in

time, t = 1, ..., T , with t = 1 denoting the first and t = T denoting the last product respectively.

Capacity is consumed in this sequence as well, and hence defining C0 = C, Ct to be the capacity

at the end of period t, we observe that Ct = Ct−1 − (ΣNi=11{χ(i)=t}πi) rt. Equation (3) enforces

the constraint that the cumulative sales over the sales horizon cannot exceed the available

capacity, and hence Ct ≥ 0, ∀ t = 1, ..., T . If capacity is endogenous, i.e., an optimization

variable, then constraint (3) can be dropped. The optimization variables are the price and

fill-rate to offer in each of these T periods, and prices are non-negative, fill-rates are between 0

and 1, that the total sales cannot exceed the available capacity.

2.2 Reformulation as a mechanism design problem

This section develops a mechanism design formulation that is equivalent to the problem specified

in (2)-(4), and enables the firm to incorporate strategic customer behavior within its revenue

optimization problem.

Sufficiency of N products: As a starting observation we show that the firm needs to offer

at most N distinct products, N being the number of customer types.

Lemma 1 Let k∗ be the optimal number of products for formulation (2)-(4). Then, k∗ ≤ N .

The above result does not preclude the case where the firm may optimally choose less than N

products, or even just one product. Hence the firm needs to segment the sales horizon of T

periods into at most N intervals such that a distinct product (price, fill-rate) combination is

offered in each interval. Since all customers are assumed to be strategic, the length (as long as

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it is non-zero) or the ordering of the intervals during which distinct products are offered does

not matter.

Reformulation as a static, product design problem: Next note that we can rewrite the

revenue in (2),ΣT

t=1 (ΣNi=11{χ(i)=t}πi) rt pt = ΣN

i=1 πi pχ(i) rχ(i), (5)

where we define p0 := 0, r0 := 0. Hence, the T period optimization problem in (2)-(4) can

be viewed as a single period problem, when we interpret the fill-rates associated with different

time periods as quality attributes of the different product variants that the firm offers to this

market of strategic customers. While customers choose the optimal time to enter the system

and purchase a product (if at all), for the firm, time does not explicitly enter the problem. The

firm needs to compute the optimal prices and fill-rates as if it were a single period problem and

all the customers arrive, purchase (if at all), and depart in the same period. This mapping is

illustrated in Figure 1.

1 2 . . .

.

.

.

1

2

N

.. . .

.

.

ProductVariants

Time

?

?T

Figure 1: In model (a), customers strategize over the timing of their purchases. Model (b) interpretseach time period as a product variant, and customers strategize over which variant to choose, if any.Also, a solution to model (a) can be mapped to a solution to model (b), and vice-versa.

The above observation allows us to reformulate the dynamic pricing problem as a static

mechanism design problem. Customers arrive and observe the product menu offered by the

firm, and make their choices accordingly. Each customer is characterized by its type designation,

which is private information, i.e., not directly observed by the firm. The firm’s problem is to

design the optimal product menu so as to maximize its profitability.

To begin with, one can restrict the firm’s optimization problem to so called “direct mecha-

nisms”, wherein the firm designs a payment and product allocation policy (“the mechanism”),

under which the customers choose to truthfully self-report their type, as described in Myerson

[11]. Essentially, in order to elicit this private type information, the mechanism is designed in

a way such that the customer is allocated the product variant that she/he would have selected

on her/his own. Following Lemma 1, which ensures that we need to offer at most N distinct

8

products, the resulting problem can be formulated as follows:

max ΣNi=1 pi πi ri (6)

s.t. (vi − pi)γi ri ≥ (vi − pj)γi rj , ∀j 6= i, (7)

(vi − pi) ri ≥ 0, i = 1, 2, ..., N, (8)

ΣNi=1 πi ri ≤ C, (9)

0 ≤ pi, 0 ≤ ri ≤ 1, i = 1, 2, ..., N. (10)

Equations (6) assumes, without loss of generality, that customer type i buys product i, i.e.,

χ(i) = i. Equations (7) are the Incentive Compatibility (IC) conditions, enforcing that customer

type i (at least weakly) prefers product i over all other products offered by the firm. Equations

(8) are the Individual Rationality (IR) conditions, enforcing that customer type i buys from the

firm only if the resulting consumer surplus is non-negative. Equation (9) enforces the capacity

constraint, and can be removed if capacity is endogenous. In what follows we will analyze

(6)-(10) assuming that capacity C is exogenous, with the understanding that the endogenous

capacity case can be solved by dropping the capacity constraint (or by setting C = ∞). In

the optimal solution, some products can be the same, thereby allowing less that N distinct

products to be offered. In our solution, ri = 0 implies that customer type i is not offered a

product. The above discussion leads to the following theorem, which we state without proof.

Theorem 1 The problem (2)-(4) is equivalent to the product design problem (6)-(10) in the

sense that both formulations lead to the same optimal solution.

Translation of product design solution to a dynamic policy: Given a solution to (6)-

(10), a solution to (2)-(4) can be obtained by assigning to each unique variant (price, fill-rate

combination), an interval of time over which it will be applied. For example, if the solution

(6)-(10) involves offering k distinct products, then one possible assignment is to offer the k

variants in k disjoint intervals, each of length bT/kc. Since customers are strategic, arrive at

the beginning of the time horizon, and demand is deterministic, the order in which different

variants are offered, or the duration of time for which they are offered, does not matter in our

stylized model. In a richer model, it might be optimal to offer products in a certain order, e.g.,

in increasing order of prices, as in Su [14]. The mechanism design formulation can incorporate

other model attributes, such as time-discounting, myopic behavior, etc. that would force the

solution to “define” the sequencing of product variants over time.

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3 Analysis of the mechanism design problem

The mechanism design formulation (6) - (10) allows us to deduce several structural properties

of the optimal solution. In what follows, without loss of generality, we will assume that χ(i) =

i, i = 1, ..., N , whenever we consider an N product solution.

Lemma 2 (Monotonicity of prices and fill-rates) At the optimal solution for (6)-(10),

p1 ≥ p2 ≥ ... ≥ pN and r1 ≥ r2 ≥ ... ≥ rN . Moreover, if for some i 6= j, pi > pj then ri > rj,

and if pi = pj, then ri = rj.

The next lemma shows that it suffices for type i customers (buying product i) to only check the

IC constraints for products intended for types i− 1 and i + 1, if any, thus reducing the number

of IC constraints from N(N − 1) to 2(N − 1).

Lemma 3 (Transitivity of IC constraints) The IC conditions in (7) are equivalent to:

(vi − pi)γi ri ≥ (vi − pi+1)γi ri+1, i = 1, 2, ...N − 1, (11)

(vi − pi)γi ri ≥ (vi − pi−1)γi ri−1, i = 2, ...N. (12)

We will refer to (11) and (12) as the downstream and upstream IC constraints, and from now

on replace (6)-(10) by (6), (8)-(12). Lemma 4 shows that products offered by the firm partition

the customer types into contiguous sets, so that if customer types i − 1 and i + 1 buy the

same product l, then customer type i also buys product l. This property will be exploited in

subsequent computational algorithms.

Lemma 4 (Contiguous Partitioning) Suppose the firm offers k ≤ N distinct products with

p1 > p2 > ... > pk and r1 > r2 > ... > rk, and such that each generates non-zero demand.

Then, these products partition the customer types into contiguous sets {1, ..., i1}, {i1 +1, ..., i2},..., {ik−1 + 1, ..., ik}, buying product 1, 2, ..., k, respectively, 1 ≤ i1 < i2 < ... < ik ≤ N , and

customer types {ik + 1, ..., N}, if any, not buying from the firm. In addition,

a) r1 = min(

1, C

Σi1l=1πl

),

b) pj = vij , where j = max{1 ≤ l ≤ k | rl > 0} in the optimal solution.

In what follows, we will assume that whenever k ≤ N products are offered, they partition the

customer types as in Lemma 4, i.e., χ(1) = ... = χ(i1) = 1, χ(i1 + 1) = ... = χ(i2) = 2, ...,

χ(ik−1 + 1) = ... = χ(ik) = k, and χ(ik + 1) = χ(ik + 2) = ... = 0, if any.

Using Lemma 4, we can characterize the optimal one product solution.

Corollary 1 (One product solution) The optimal one product solution is p∗ = vi, r∗ =

min(1, C

Σik=1πk

)where i = argmaxmin(C, Σj

k=1πk)vj. Moreover, if C ≤ π1, then the globally

optimal solution is to offer a single product to type 1 customers with p1 = v1 and r1 = Cπ1

.

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To avoid trivial solutions, hereafter we will assume that C > π1. The next result shows that

the upstream IC constraints for types 2, ..., N can be dropped.

Proposition 1 Formulation (6), (8)-(12) has the same (optimal) solution with (6), (8)-(11).

The proof of Proposition 1 establishes that constraint (11) is tight in the optimal solution, and

so we can use it to express the optimal fill-rates in terms of the optimal prices.

Corollary 2 a) A price vector p defines a partitioning of the customer types, specifically, p1 =

p2 = ... = pi1 > pi1+1 = ... = pi2 > ... > pik−1+1 = ... = pik , pij ≤ vij , j = 1, ..., k − 1, pk = vik ,

partitions the customer types as described in Lemma 4.

b) Fixing the price vector as above, the optimal fill-rates for j = 1, ..., k, are given as follows:

rj = min

max

C − Σj−1

l=1 (Σilm=il−1+1πm) rl

Σijm=ij−1+1πm

, 0

, Πj−1

l=1

(vil − pl

vil − pl+1

)γil

. (13)

The above observation holds at the optimal solution. Also, given an exogenously fixed price

vector p satisfying the monotonicity condition in Corollary 2, problem (6), (8)-(11) is solvable

in closed form. Formulation (6), (8)-(11) also leads to the following corollary, which relates the

optimal revenue with risk-neutral customers to optimal revenue with risk-averse customers.

Corollary 3 Let R(γ) be the optimal revenue achieved for (6), (8)-(11) with risk-aversion

vector γ. Let 1 denote the N -vector of ones. Then R(γ) ≥ R(1), ∀γ ≤ 1, where R(1) denotes

the optimal revenue achieved for (6), (8)-(11), for risk-neutral customers.

4 Computations

Problem (6), (8)-(11) appears hard, in part due to the bilinear objective, but mostly due to the

non-convex nature of the constraint (11) for γ < 1. We next discuss two special cases where

it can be solved efficiently, and then relate them for our key computational and managerial

insight regarding the near optimality of two-product strategies in low risk-aversion settings.

4.1 Risk-neutral case

When customers are risk-neutral, objective (6) and constraint (11) can be simplified through

appropriate variable substitutions to lead into an equivalent LP formulation.

Proposition 2 If γi = 1, i = 1, 2, ..., N , then (6), (8)-(11) can be reformulated as the following

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LP: choose yi, zi, i = 1, ..., N to

max ΣNi=1 πi zi (14)

s.t. zi − zi+1 = vi yi, i = 1, 2, ..., N − 1, (15)

ΣNi=1 (ΣN

i=i yi) πi ≤ C, (16)

vi ΣNi=i yi ≥ zi, (17)

0 ≤ zi, 0 ≤ ΣNi=i yi ≤ 1, (18)

where zi := piri and yi := ri − ri+1, i = 1, ..., N , yN+1 := 0.

This LP gives us zi and yi as solution. However, yN = rN , and yi = ri− ri+1, so we can obtain

ri, i = 1, ..., N . Next the relation zi = piri gives us the value of pi, i = 1, ..., N . Proposition 2

implies that the firm’s problem is easy to solve in the case of risk-neutral customers. Our next

proposition shows that there exists a solution to (14)-(18) that involves offering at most two

distinct products, irrespective of the customer valuation distribution and available capacity.

Proposition 3 If γi = 1, i = 1, ..., N , then the optimal number of products to offer to risk-

neutral customers, k, is at most 2. In particular, k = 1 if the capacity constraint is slack in the

optimal solution, and k = 2 if the capacity constraint is tight in the optimal solution.

The proof of the above proposition also leads to the following corollary, which states that under

our assumption that C > π1, the highest fill-rate is always equal to 1 in the optimal risk-neutral

solution irrespective of the available capacity.

Corollary 4 For risk-neutral customers, under π1 ≤ C, it is never optimal to set r1 < 1.

4.2 Two product case (k = 2)

For administrative reasons (“menu” costs associated with offering new products) or branding

considerations (the firm may not want to ration in multiple periods, so that customers that are

rationed our in an earlier period do not find the product to be available in a later period), the

seller may want to offer a small number of products, typically two. This section solves that

problem. The two products effectively partition the N customer types into 3 segments. The

first segment of customers from types 1 to i1 buys product 1, the second segment of customers

from types i1 + 1 to i2 buys product 2, and the remaining customer types, if any, do not buy

from the firm. Algorithm 1 outlines how to solve this problem efficiently in O(N2) time (the

proof is presented in Appendix B).

From Lemma 1, we know that at most two distinct products need to be offered in a market

with two customer types, and so it follows that we can solve the two customer type problem

12

Algorithm 1 To calculate two distinct product solutionR∗ = 0for i1 = 1 to N − 1 do

for i2 = i1 + 1 to N doRi1,i2 = 0if γi1 < 1 then

if C ≥ (Σi1l=1πl) + (Σi2

l=i1+1πl)(Σ

i2l=i1+1πlvi2γi1

(Σi1l=1πl)(vi1−vi2 )

)

γi11−γi1 && (Σi1

l=1πl)(vi1 − vi2) > (Σi2l=i1+1πl)vi2γi1

then

Ri1,i2 = (Σi1l=1πl)v1 + (

(Σi2l=i1+1πl)vi2γi1

(Σi1l=1πl)

γi1 (vi1−vi2 )γi1

)1

1−γi1 ( 1γi1

− 1)

else if C < (Σi1l=1πl) + (Σi2

l=i1+1πl)((Σ

i2l=i1+1πl)vi2γi1

(Σi1l=1πl)(vi1−vi2 )

)

γi11−γi1 && C < (Σi1

l=1πl) + (Σi2l=i1+1πl) then

Ri1,i2 = (Σi1l=1πl)vi1 − (Σi1

l=1πl)(vi1 − vi2)(C−Σ

i1l=1πl

Σi2l=i1+1πl

)1

γi1 + (Σi2l=i1+1πl)vi2(

C−Σi1l=1πl

Σi2l=i1+1πl

)

else if γi1 = 1 thenif C ≥ (Σi1

l=1πl) + (Σi2l=i1+1πl) && (Σi1

l=1πl)(vi1 − vi2) == (Σi2l=i1+1πl)vi2 then

Ri1,i2 = (Σi1l=1πl)v1

else if C < (Σi1l=1πl) + (Σi2

l=i1+1πl) && (Σi1l=1πl)(vi1 − vi2) < (Σi2

l=i1+1πl)vi2 then

Ri1,i2 = (Σi1l=1πl)vi1 − (Σi1

l=1πl)(vi1 − vi2)(C−Σ

i1l=1πl

Σi2l=i1+1πl

) + (Σi2l=i1+1πl)vi2(

C−Σi1l=1πl

Σi2l=i1+1πl

)

if R∗ < Ri1,i2 thenR∗ = Ri1,i2

if R∗ = 0 thenNot optimal to offer two distinct products

as a special case of the two product problem. The two product solution provides a lower

bound to the optimal revenues attainable in problem (6), (8)-(11). Since the general problem

is hard to solve, this provides a heuristic solution to the problem. The next section show that

it is asymptotically optimal in settings with low risk-aversion, and its overall effectiveness is

evaluated numerically in section 6.

5 Low risk-aversion: offering two products is near-optimal

We now focus on the setting where customers have low risk-aversion, i. e., γ close to 1. To that

end, we will rewrite γi = 1− xi, where xi := 1−γi

1−γ1and x1 = 1 ≥ x2 ≥ ... ≥ xN ≥ 0. We assume

that γ1 < 1, else the problem involves risk-neutral customers only. We will consider a sequence

of problems indexed by n, where the nth problem is characterized by the risk-aversion parameter

vector γn, given by γni = 1 − xi

n for i = 1, ..., N . When n = 11−γ1

, we “recover” the original

model parameters, or in other words, the element in the above sequence that corresponds to

n = 11−γ1

is exactly the one we started with. We are interested in the case where γ1 ↑ 1. 2

2It is also possible to consider the case where γn ↓ 0, wherein the asymptotically optimal solution can becharacterized as follows. Let i∗ = min{i | Σi

l=1πl > C}. Then, pni → vi, i < i∗, pn

i = vi, i ≥ i∗, rni → 1, i < i∗,

rni∗ =

C−Σi∗−1l=1 πl

πi∗, rn

i = 0, i > i∗, rn1 > rn

2 > ... > rni∗ > 0, and the optimal revenue converges to revenue

achievable with myopic customers. This case corresponds to extremely high risk-aversion, and we do not analyzeit in detail.

13

We will denote the optimal solution to the problem with γn as the risk-aversion parameter

vector as (pn, rn), and the optimal solution to the risk-neutral problem as (p, r). Following

Corollary 2, given price-vector pn, the corresponding fill-rate vector rn is uniquely determined.

So, in what follows, we will often abbreviate (pn, rn) to pn and (p, r) to p. We will also refer to

the problem with γn as the risk-aversion parameter vector as Pn and the risk-neutral problem

as P. We will denote the feasible set, given by equations (8)-(12), for Pn as Sn and the feasible

set for P as S.

Asymptotic optimality of risk-neutral solution: We will make the following assumption

in the subsequent analysis.

Assumption 1: The risk-neutral solution (p, r) is unique.

Proposition 4 Under assumption 1, for any convergent subsequence {pnk}, pnk → p, and

R(γnk) → R(1), as nk ↑ ∞.

The proof of Proposition 4 also shows that for n sufficiently large, the optimal risk-neutral

solution (p, r) is feasible for the problem with risk-averse customers, Sn, and that the optimal

solution pn is “close” to a feasible risk-neutral solution p′ ∈ S.

Perturbations around (p, r): With this knowledge, we will look at the perturbed solution to

P as a candidate optimal solution for Pn for n sufficiently large. Specifically, for Pn, we will

consider solutions of the form (p + δn, r + ρn) for the risk-averse problem, where δn := δn , and

ρn := ρn , such that δn = pn − p, and ρn = rn − r. Suppose the optimal risk-neutral solution

partitions involves offering k ≤ N distinct products, where the products partition the customer

types as in Lemma 4. If ik < N , define j := ik + 1, and set ri = 0, pi = vi, i ≥ j. Then, the

revenue-maximization problem Pn can be written as follows.

max ΣNi=1πi(pi + δn

i )(ri + ρni ) (19)

s.t. ΣNi=1πi(r + ρn

i ) ≤ C, (20)

(vi − pi − δni )γi(ri + ρn

i ) ≥ (vi − pi+1 − δni+1)

γi(ri+1 + ρni+1), i = 1, ..., N − 1, (21)

(vi − pi − δni )(ri + ρn

i ) ≥ 0, i = 1, ..., N − 1, (22)

δni ≥ δn

i+1, i /∈ {i1, i2, ..., ik}, i < ik, ρni ≥ ρn

i+1, i /∈ {i1, i2, ..., ik}, (23)

δnik≤ 0, ρn

i ≤ 0, i = 1, ..., i1, (24)

δni ≤ 0, ρn

i ≥ 0, i ≥ j. (25)

Equations (19) and (20) represent the objective and the capacity constraint, respectively, for

the problem Pn. Equation (21) is the downstream IC constraint. Equation (22) ensures that

the IR condition is satisfied. Equation (23) ensures that prices and fill-rates are monotonically

non-increasing with respect to customer types. For n sufficiently large, we need to enforce this

condition only for indices i /∈ {i1, i2, ..., ik}. Equation (24) ensures that pnik

cannot increase from

14

the optimal price pik= vik for the risk-neutral case, and similarly that the fill-rate rn

1 cannot

increase from the optimal fill-rate r1 = 1 for the risk-neutral case (Following Corollary 4 and our

assumption that π1 ≤ C, the optimal solution involves setting r1 = 1). Finally, equation (25)

ensures that prices and fill-rates for types that were not being sold a product in the risk-neutral

case (indices i ≥ j), if any, are all non-negative.

Characterization of the optimal perturbation around (p, r): Since γn = 1− xn , in what

follows, we will use Taylor expansion and focus on the second-order terms. This will lead to a

LP in terms of x, δ and ρ. This would imply that the price and rationing risk “corrections”

needed because customers are not risk-neutral are captured through a solution to a LP. We

proceed to derive this LP as follows.

The objective in (19) can be re-written as ΣNi=1(πipiri + πipiρ

ni + πiδ

ni ri + πiρ

ni δn

i ), wherein

we note that first term is objective is a constant, while the last term is O( 1n2 )3. Similarly,

equation (20) can be re-written as ΣNi=1(πiri + πiρ

ni ) ≤ C. If the capacity constraint for P

is slack at the optimal solution (p, r), then this constraint can be dropped (since as n grows

large, the first-order term will dominate). If not, since the optimal risk-neutral solution was

capacitated, it can be re-written as ΣNi=1πiρ

ni ≤ 0. Finally, using Taylor expansion, the IC

constraint (21) can be written as

(vi − pi)ρni − δn

i ri − (vi − pi)ri(1− γni ) log(vi − pi) + O

(1/n2

) ≥(vi − pi+1)ρ

ni+1 − δn

i+1ri+1 − (vi − pi+1)ri+1(1− γni ) log(vi − pi+1) + O

(1/n2

),

(26)

where we used that (vi − pi+1)ri+1 = (vi − pi)ri, which follows from the tightness of the

downstream IC condition in the optimal solution to P, as shown in Proposition 1. We will

substitute this constraint by the following constraint.

(vi − pi)ρni − δn

i ri − (vi − pi)ri(1− γni ) log(vi − pi)+ ≥

(vi − pi+1)ρni+1 − δn

i+1ri+1 − (vi − pi+1)ri+1(1− γni ) log(vi − pi+1) + εi,

(27)

where εi > 0, if i ∈ {i1, i2, ..., ik−1}, εi = 0 otherwise. For n sufficiently large, feasibility of

constraint (27) implies feasibility of constraint (26). In what follows we will use the following

notation.

ui,i = vi − pi, wi,i = (vi − pi)ri log(vi − pi), (28)

ui,i+1 = vi − pi+1, wi,i+1 = (vi − pi)ri log(vi − pi+1). (29)

The above discussion leads to the following first-order optimization problem.

max ΣNi=1(πipiρi + πiriδi) (30)

s.t. ΣNi=1πiρi ≤ 0, (31)

3g(x) = O(f(x)) denotes that limx↓0g(x)f(x)

= c < ∞

15

ρi ≤ 0, i = 1, ..., i1, (32)

δik ≤ 0, (33)

δi ≥ δi+1, i /∈ {i1, i2, ..., ik}, i < ik, ρi ≥ ρi+1, i /∈ {i1, i2, ..., ik}, (34)

δi ≤ 0, ρi ≥ 0, i ≥ j, (35)

ui,iρi + δi+1ri+1 + wi,i+1xi ≥ ui,i+1ρi+1 + δiri + wi,ixi + εi, i = 1, ..., N − 1. (36)

Analyzing the dual of problem (30)-(36), verifying its feasibility and using strong duality for

LPs leads to the following proposition.

Proposition 5 The problem (30)-(36) is feasible and has a finite solution. Let k denote the

optimal number of products to offer for the risk-neutral problem S. Then,

i) if k = 1, ρi = δi = 0, i = 1, ..., N ,

ii) if k = 2, ρi = 0, i = 1, ..., N , δ1 = δ2 = ... = δi1 = (wi1,i1+1 − wi1,i1)xi1 − εi1, δi1+1 = ... =

δN = 0, for εi1 > 0 sufficiently small.

Proposition 5 implies that as γ ↑ 1, it becomes asymptotically optimal to offer at most two

products. Following Lemma 1 and algorithm 1, the optimal one product and the optimal

two distinct product solution can be computed efficiently, and together with Proposition 5, this

implies that we can solve for the optimal prices and fill-rates for the low risk-aversion case. This

section has methodically showed when and why is a two product solution, i. e., offering the

product at two price points at different fill-rates, near optimal. This allows justification for this

practical heuristic, and allows one to circumvent the intractability of the general formulation (6),

(8)-(11). It also lends credibility to numerous papers that have restricted attention to two

product models but without any theoretical justification, highlighting the conditions under

which it is suitable to do so.

6 Numerical Results

We conclude with some numerical results that study the performance of the two-product heuris-

tic. We consider a market with seven customer types (N = 7), with uniform valuations

vi = 8 − i, i = 1, ..., 7. Type i population-size is sampled from a normal distribution with

mean µi, and variance σ2i , where µ1 = 1, µ2 = 3, µ3 = 2, µ4 = 1, µ5 = 3.5, µ6 = 5, µ7 = 3,

and σi = 0.2µi. Correlation is assumed to be 0, though it can be easily added. For σi = 0, i =

1, ..., 7, this corresponds to a bimodal distribution of customer valuations. Other valuation

distributions, e.g., uniform, geometric, lead to similar results and are therefore not included.

Capacity is fixed at 1, while the capacity to market-size ratio CΣN

i=1πivaries between (0, 1] and is

also a simulation input. The risk-aversion parameter varies between (0, 1] and is a simulation

16

90%

92%

94%

96%

98%

100%

0.0 0.2 0.4 0.6 0.8 1.0

Risk-aversion parameter

Tw

o p

rod

uct / O

ptim

al R

eve

nu

e

Cap/MktSize=1.00

Cap/MktSize=0.75

Cap/MktSize=0.50

(a) Ratio of two-product to optimal revenue

0%

20%

40%

60%

80%

100%

0.0 0.2 0.4 0.6 0.8 1.0

Risk-aversion parameter

% c

ase

s t

wo

pro

du

cts

op

tim

al Cap/MktSize=1.00

Cap/MktSize=0.75

Cap/MktSize=0.50

(b) % scenarios where at most two products suffice

1

2

3

4

5

6

0.0 0.2 0.4 0.6 0.8 1.0

Risk-aversion parameter

Op

tim

al #

pro

du

cts

to

off

er

Capacity/MktSize=1.00

Capacity/MktSize=0.75

Capacity/MktSize=0.50

(c) Optimal number of products to offer

2.0

2.5

3.0

3.5

4.0

4.5

0.0 0.2 0.4 0.6 0.8 1.0

Risk-aversion parameter

Revenue

One-product Two product

k-product Myopic customers

(d) Revenue at Capacity/Market-size = 0.75

Figure 2: This figure shows how the two-product solution compares to the k-product solution as afunction of risk-aversion for different capacity to market-size ratios. Figure (a) shows that the two-product solution approaches the optimal solution as risk-aversion parameter approaches 1. Figures (b)and (c) show how the proportion of cases where it is optimal to offer two products, and the optimalnumber of products to offer, respectively, varies with the risk-aversion parameter. These results areaveraged over 50 demand scenarios. Figure (d) examines one such demand scenario in detail and showsthat the k-product revenue decreases monotonically and approaches the two-product revenue as risk-aversion approaches one. The maximum revenue is obtained with myopic customers, followed by thek-product, two-product and one-product revenue.

input. γ2, ..., γN are assumed to be order-statistics of random samples drawn uniformly from

the interval [γ1, 1], so that γ1 ≤ γ2 ≤ ... ≤ γN ≤ 1. Given γ1 and a fixed capacity to market-

size ratio, 50 scenarios are generated, wherein for each scenario, the risk-aversion parameters

γ2, ..., γN are generated randomly as described above, and the customer population size π is

sampled from a normal distribution with the parameters given above. Negative demand, if any,

is truncated to zero, and the resulting customer population sizes are scaled proportionately to

achieve the given capacity to market-size ratio. For each scenario, the optimal one-product,

two-product and k-product solutions are computed. These are averaged over scenarios to obtain

results corresponding to a data point in Figure 2.

Figure 2 (a) shows how the two-product revenue compares with the k-product revenue for

different capacity to market-size ratios as γ1 varies between zero and one. We observe that as γ1

17

approaches one, the two-product revenue approaches the k-product revenue. Even when γ1 is

small and not close to 1, the two-product revenue achieves within 8% of the k-product revenues

on average, therefore serving as a useful approximation and lower bound. (As described in

Footnote 1 in section 5, as γ ↓ 0, the optimal number of products may grow large.) Figure 2

(b) shows how the fraction of scenarios where it suffices to offer at most two products varies

with γ1. We observe that this fraction is non-monotonic, but the overall trend suggests that it

increases as γ1 increases. It equals one in the limit of risk-neutral customers, but for other risk-

aversion values, it can be much less than one. From Figure 2 (a), we know that the two product

revenue is close to the k-product revenue, and together this implies that even though the two-

product solution might be suboptimal in a large fraction of cases, the two-product solution is

close to the k-product solution and hence the suboptimality gap is small. Figure 2 (c) shows

the optimal number of products to offer as a function of γ1. Again, while non-monotonic, the

overall trend suggests that this number decreases as γ1 increases. For the risk-neutral case,

this number lies between one and two, consistent with our result that at most two products

need to be offered in this case. Figure 2 (d) shows the one-product, two-product and k-product

revenue as a function of γ1 for the case where capacity to market-size ratio is fixed at 0.75. It

also plots the optimal achievable revenue if the customers were all myopic. We observe that

the highest revenue is achieved when customers are myopic, followed by the k-product solution,

the two-product solution, and the one-product solution, respectively. Both the revenue with

myopic customers and the one-product revenue do not depend on customer risk-aversion. The

k-product revenue dominates the two-product revenue, and approaches it as γ1 approaches one.

Also, the k-product revenue decreases as γ1 increases, and as expected, exceeds the revenue

with risk-neutral customers.

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Appendix A

Proof of Lemma 1 From (1), all type i customers select the same product variant, χ(i, p, r).

Since there are N types, there can be at most N distinct products that generate non-zero

demand. Hence, it suffices to offer at most N distinct products. ¤

Proof of Lemma 2 We first show that higher prices are associated with higher fill-rates. IC

conditions for types i and j (that prefer product i and j, respectively) imply that,(

vi−pi

vi−pj

)γi ≥ rj

ri

19

and(

vj−pj

vj−pi

)γj ≥ rirj

, respectively. There are three cases to consider. 1) Suppose pi = pj : then,

from the IC condition we obtain ri = rj . 2) pi > pj : then, the IC condition for type i implies

that rj

ri≤

(vi−pi

vi−pj

)γi

< 1. So, rj < ri. 3) pi < pj : this case is symmetric to 2). Similarly, one

can verify that higher fill-rates are associated with higher prices.

Next we show that “higher” types prefer higher priced products. Suppose that there exist

i < j, such that pi < pj . IC conditions for customer types i and j imply that(

vi−pi

vi−pj

)γi ≥rj

ri≥

(vj−pi

vj−pj

)γj

. Also, pi < pj and vi > vj imply that 1 < vi−pivi−pj

<vj−pi

vj−pj, and since γi ≤ γj ,(

vi−pi

vi−pj

)γi

<(

vj−pi

vj−pj

)γj

. The latter implies that the IC conditions cannot hold, and this leads

to a contradiction. ¤

Proof of Lemma 3 The proof comprises of two parts. First we show that customer type i−1

would rather make the choice made by type i than by i + 1. Then we show that customer type

i + 1 would rather make the choice made by type i than by i− 1. Together they imply that IC

conditions are transitive.

Step 1 : We will show that if type i chooses product i over product i + 1, then type i − 1 will

also choose product i over product i + 1. Following Lemma 2, pi−1 ≥ pi ≥ pi+1. Consequently,

1 ≥ vi−1−pi

vi−1−pi+1≥ vi−pi

vi−pi+1, and so γi−1 ≤ γi implies that

(vi−1−pi

vi−1−pi+1

)γi−1 ≥(

vi−pi

vi−pi+1

)γi

. The IC

condition that guarantees that type i customers choose product i over product i + 1 implies

that(

vi−pi

vi−pi+1

)γi ≥ ri+1

ri, thereby leading to the desired inequality.

Step 2 : Similarly, one can show that if type i chooses product i over product i− 1, then type

i + 1 will also choose product i over product i− 1 (details are omitted). ¤

Proof of Lemma 4 Suppose the seller decides to offer only k ≤ N products. Then, we will

show that if customer types i−1 and i+1 choose product l, then customer type i chooses product

l, this would imply that offered products partition the customer types into contiguous sets. IC

conditions for type i−1 and type i+1 customers imply that, for any m 6= l, ( vi−1−pl

vi−1−pm)γi−1 ≥ rm

rl

and(

vi+1−pl

vi+1−pm

)γi+1 ≥ rmrl

. Consider all products m s.t. vi > pm > pl. Then, 1 < vi−1−pl

vi−1−pm<

vi−plvi−pm

. Next γi−1 ≤ γi implies that(

vi−plvi−pm

)γi

> rmrl

, and so type i customers also prefer

product l over all products m with vi > pm > pl. Now consider all products m s.t. pm < pl.

Then, 1 > vi−plvi−pm

> vi+1−pl

vi+1−pm. Next γi ≤ γi+1 implies that 1 >

(vi−plvi−pm

)γi

> rmrl

, so that type i

customers also prefer product l over all products m with pm < pl.

Parts a) and b) of the Lemma proceed as follows. a) Since p1 ≥ p2 ≥ ... ≥ pk follow-

ing Lemma 2, setting r1 to the highest possible value is optimal. This is min(

1, C

Σi1l=i0+1πl

).

b) Suppose in a given optimal solution (p, r), pj < vij where j = max{1 ≤ l ≤ k|rl > 0}. Define

a new price vector as follows: p′j = vij , p′l = vil − (vil − p′l+1)(rl+1

rl)

1γil , 1 ≤ l < j. Observe that

20

vil ≥ p′l ≥ pl, 1 ≤ l ≤ j, and none of the customer types that were buying a product switch

classes or discontinue to buy the product. The solution (p′, r) is also incentive-compatible and

feasible, and results in a greater revenue. Hence pj < vij cannot hold in an optimal solution. ¤

Proof of Corollary 1 From Lemma 4, we know that r∗ = r1 = min(1, C

Σik=1πi

), where

customer types 1, .., i buy product 1. Also the optimal price to offer the product at when up to

type i are being served the first product is vi, and hence we can search for the optimal value of

i by evaluating the revenue at each of the N possible price points v1, v2, ..., vN . ¤

Proof of Proposition 1 Suppose the optimal solution to formulation (6), (8)-(12) involves of-

fering k ≤ N distinct products, where the products partition the customer types as in Lemma 4.

Given this partitioning, it suffices to impose downstream IC constraints for types i1, i2, ..., ik−1

and upstream IC constraints for types i1 +1, i2 +1, ..., ik−1 +1. Just as in the case of individual

customer classes, transitivity across groups also holds.

We next determine the necessary conditions that the optimal prices and the fill-rates must

satisfy. Since k distinct products are offered, 0 < vik < pj < vij , j = 1, ..., k − 1, rj > 0, j =

1, ..., k, and the Lagrange multipliers with the associated bounding constraints vi ≥ pi, i =

1, ..., ik− 1, pi ≥ 0, i = 1, ..., ik, 1 ≥ ri, i = i1 +1, ..., ik, ri > 0, i = 1, ..., ik are zero. Moreover,

since the solution is optimal, pk = vik and r1 = 1, so that we have 2k - 2 optimization variables.

We can write the Lagrangian as follows (fulfilment of the constraint qualification condition is

shown in Appendix B):

L = Σkj=1(Σ

ijl=ij−1+1πl)pjrj + Σk−1

j=1µj

((vij − pj)

γij rj − (vij − pj+1)γij rj+1

)

+ Σk−1j=1ζj

((vij+1 − pj+1)

γij+1rj+1 − (vij+1 − pj)γij+1rj

)+ λ(C − Σk

j=1Σijl=ij−1

πlrj).(37)

Note that µjζj = 0, since exactly one of the respective constraints is tight; otherwise we

can increase revenues by changing the price or the fill-rate. Differentiating with respect to p1,

we obtain

∂L

∂p1= Σi1

l=1πl − µ1γi1(vi1 − p1)γi1−1 + ζ1γi1+1(vi1+1 − p1)γi1+1−1 = 0,

implying that µ1 > 0, ζ1 = 0. Differentiating with respect to pu, we get that

∂L

∂pu= (Σiu

l=iu−1+1πl)ru − µuγiu(viu − pu)γiu−1ru + µu−1γiu−1(viu−1 − pu)γiu−1−1ru

−ζu−1γiu−1+1(viu−1+1 − pu)γiu−1+1−1ru + ζuγiu+1(viu+1 − pu)γiu+1−1ru = 0.

Now using the induction hypothesis that µu−1 > 0, ζu−1 = 0, we find that µu > 0, ζu = 0. This

implies that all the downstream constraints are tight, while all the upstream constraints are

slack. Hence, given any partitioning, we can drop the upstream constraints. Moreover, we can

set the downstream constraints to be tight. Since, the choice of partitioning does not matter,

21

this holds for all partitions, and in particular the optimal partition, and hence formulation (6),

(8)-(11) leads to the same optimal solution as formulation (6), (8)-(12).

Proof of Corollary 2 a) This follows directly from Lemma 4. b) The expression for r follows

from the tightness of constraint (11) in formulation (6), (8)-(11), the second part of Lemma 4,

and the non-negativity of fill-rates. ¤

Proof of Corollary 3 Note that the risk-aversion parameter enters formulation (6), (8)-(11)

only via constraints (11). Next, denote the optimal solution to formulation (6), (8)-(11) when

customers are risk-neutral by (p, r). Then(

vi−pivi−pi+1

)γi ≥(

vi−pivi−pi+1

)≥ ri+1

ri, and hence (p, r)

is feasible for formulation (6), (8)-(11) with γ as risk-aversion parameter vector. Hence the

revenue with (p, r) serves as a lower bound for the optimal revenue to the risk-averse problem.

(Note that (p, r) might not be feasible for formulation (6), (8)-(12) with γ as risk-aversion

parameter vector.) Similarly, one can also show that the optimal revenue with risk-aversion

parameter γ serves as a lower bound for revenue with risk-aversion parameter γ′, if γ′ ≤ γ. ¤

Proof of Proposition 2 Define zi := piri, i = 1, ..., N , and yi := ri − ri+1, i = 1, ..., N −1, yN = rN . Then the IR condition (equation (8)), the IC condition (equation (11)), and

the capacity constraint (equation (9)) can be written as viΣNl=iyl ≥ zi, zi − zi+1 ≤ viyi and

ΣNi=1(Σ

Nl=iyl)πi ≤ C, respectively, while the objective (equation (6)) becomes ΣN

i=1πizi, which

are all linear in the variables zi, yi, thereby leading to an LP. ¤

Proof of Proposition 3 Suppose customers are risk-neutral and the firm decides to offer k > 1

distinct products such that they partition customer types as in Lemma 4. Using Proposition

1, we can write the Lagrangian as

L = Σkj=1(Σ

ijl=ij−1+1πl)pjrj + Σk−1

j=1µj

((vij − pj)rj − (vij − pj+1)rj+1

)

+ λ(C − Σkj=1Σ

ijl=ij−1+1πlrj).

(38)

Using Lemma 4 and differentiating with respect to p1 yields Σi1l=1πl − µ1 = 0. Differentiating

with respect to pj gives Σijl=ij−1+1πl−µj +µj−1 = 0, j = 2, ..., k−1. Together these imply that

µj = Σijl=1πl. Differentiating with respect to rj and using the above we get µjvij − µj−1vij−1 −

λΣijl=ij−1+1πl = 0, j = 2, ..., k − 1. Using µj = Σij

l=1πl gives (Σijl=ij−1+1πl)vij − (Σij−1

l=1 πl)(vij−1 −vij )− λΣij

l=ij−1+1πl = 0. There are two cases to consider.

a) Σi1l=1πlvi1 = Σi2

l=1πlvi2 = ... = Σikl=1πlvik , when capacity is unconstrained and k > 1,

b) λ = Σi2l=1πlvi2

−Σi1l=1πlvi1

Σi2l=i1+1πl

= Σi3l=1πlvi3

−Σi2l=1πlvi2

Σi3l=i2+1πl

= ... =Σ

ikl=1πlvik

−Σik−1l=1 πlvik−1

Σikl=ik−1+1πl

≥ 0, if capacity is

scarce and k ≥ 2.

The remainder of this proof verifies (details are omitted) that the revenue in a) is given

by Σi1l=1πlvi1 , and is achieved by offering a single product at price p1 = vi1 , r1 = 1, and that

22

the revenue in case b) is given by Σi1l=1πlvi1 + λ(C − Σi1

l=1πl), and is achieved by offering two

distinct products at the following prices and fill-rates. Define u := max{j | Σijl=1πl < C}, then

p1 = viu − (viu − viu+1)C−Σiu

l=1πl

Σiu+1l=iu+1πl

, r1 = 1, p2 = viu+1 and r2 = C−Σiul=1πl

Σiu+1l=iu+1πl

. ¤

Proof of Corollary 4 Suppose r1 < 1. There are two cases to consider.

a) k > 1: Consider setting r′1 = min(

1,Σ

i1l=1πl

C

), r′j = min

(rj ,

C−Σj−1u=1(Σ

iul=iu−1+1πl)r

′i

Σijl=ij−1+1πl

), j > 1,

and increasing p′1 s.t. the downstream IC constraint (vi1 − p′1)r′1 ≥ (vi1 − p′2)r

′2 for type i1

customers is tight. Then, no type chooses to buy a different product, but the revenues strictly

increase. This leads to a contradiction. If r′1 < 1, then it implies that only one product is being

offered and hence this reduces to case b).

b) k = 1: Suppose r1 < 1. This implies that Σi1l=1πl > C and π1v1 < Cvi1 < (Σi1

l=1πl)vi1 .

Consider the following two-product offering: p′1 = v1 − (v1 − p2)r2, r′1 = 1, p′2 = vi1 , r′2 =C−π1

Σi1l=i1+1πl

. Then, the new revenue equals π1v1 + (Σi1l=1πl)vi1

−π1v1

(Σi1l=2πl)

(C − π1) > Cvi1 , again implying

that the original one product revenue was suboptimal, thereby leading to a contradiction. ¤

Proof of Proposition 4: We will first prove the following:

i) There exists M ∈ N sufficiently large s. t. (p, r) ∈ Sn,∀n ≥ M

ii) There exist p1 ∈ S, M ∈ N s. t. |pni − p1

i | < c(1− γn1 ), ∀n ≥ M , where c is a constant

i) Suppose the optimal risk-neutral solution involves offering k ≤ N distinct products, where

the products partition the customer types as in Lemma 4. Then, following Proposition 1, (p, r)

satisfies the following constraints.

(vij − pj)rj = (vij − pj+1)rj+1, j = 1, ..., k − 2,

(vij+1 − pj+1)rj+1 > (vij+1 − pj)rj , j = 2, ..., k − 1.

The feasible sets Sn and S of problems Pn and P, respectively, only differ in their IC constraints.

Hence, in order to establish claim i), it suffices to show that (p, r) satisfies the IC constraints

of Pn, for n sufficiently large. Note that

1 > (vij − pj

vij − pj+1

)γn

ij >vij − pj

vij − pj+1

=rj+1

rj,

implying that the downstream IC condition for problem Pn is satisfied by (p, r). If (vij+1−pj) ≤0, the upstream IC condition is satisfied as well, otherwise, consider the following. Define

εj , j = 2, ..., k − 1 such that

εjrj+1(vij+1 − pj) = (vij+1 − pj+1)rj+1 − (vij+1 − pj)rj > 0.

We want to show that for sufficiently large n,

(vij+1 − pj+1)γn

ij+1rj+1 ≥ (vij+1 − pj)γn

ij+1rj ,

23

⇔(

vij+1 − pj+1

vij+1 − pj

)γnij+1

≥ rj

rj+1=

(vij+1 − pj+1

vij+1 − pj

)− εj .

Substituting γnij+1 = 1−xij+1 and using the Taylor expansion, this condition can be written as(

vij+1 − pj+1

vij+1 − pj

)− c1xij+1 + O((xij+1)2) ≥

(vij+1 − pj+1

vij+1 − pj

)− εj ,

where c1 is a constant. The latter is satisfied if c1xij+1 ≤ εj +O((xij+1)2), from which it follows

that for any εj > 0, there exists a M sufficiently large such that for all n ≥ M , (p, r) satisfies

both upstream and downstream IC constraints, and this completes the proof of claim i).

ii) Suppose the optimal solution (pn, rn) involves offering k ≤ N distinct products, where

the products partition the customer types as in Lemma 4. Since (pn, rn) is the optimal solution

for Pn, it satisfies the following constraints

(vij − pnj )

γnij rn

j = (vij − pnj+1)

γnij rn

j+1, j = 1, ..., k − 1, (39)

(vij+1 − pnj+1)

γnij+1rn

j+1 > (vij+1 − pnj )

γnij+1rn

j , j = 2, ..., k − 1. (40)

We will construct a new solution (p′, r′) ∈ S, where r′ = rn, and

(vij − p′j)r′j = (vij − p′j+1)r

′j+1 [ =⇒ (vij+1 − p′j+1)r

′j+1 > (vij+1 − p′j)r

′j ],

such that |pni −pi| < c(1−γn

i ), for n sufficiently large and some constant c. Consider the k−1th

downstream IC constraint. From (39), it follows that

(vik−1− pn

k−1)rnk−1 < (vik−1

− pnk)rn

k .

Set p′k = pnk , p′k−1 = pn

k−1 − εk−1, εk−1 > 0, s.t.

(vik−1− p′k−1)r

′k−1 = (vik−1

− p′k)r′k.

This requires that

εk−1 = (vik−1− pn

k)r′k

r′k−1

− (vik−1− pn

k−1),

= (vik−1− pn

k)(

vik−1− pn

k−1

vik−1− pn

k

)γnik−1 − (vik−1

− pnk−1),

= ck−1(1− γnik−1

) + O((1− γnik−1

)2),

following a Taylor expansion, where ck−1 is a constant. Note that εk−1 > 0 since pk−1 > pk.

Next consider the k − 2th downstream IC constraint. Set p′k−2 = pnk−2 − εk−2, εk−2 > 0, s.t.

(vik−1− p′k−2)r

′k−2 = (vik−1

− p′k−1)r′k−1.

This requires that

εk−2 = (vik−2− p′k−1)

r′k−1

r′k−2

− (vik−2− pn

k−2),

24

= (vik−2− p′k−1)

(vik−2

− pnk−2

vik−2− pn

k−1

)γnik−2

− (vik−2− pn

k−2),

= (vik−2− pn

k−1 + εk−1)

(vik−2

− pnk−2

vik−2− pn

k−1

)γnik−2

− (vik−2− pn

k−2),

= c1k−2(1− γn

ik−2) + c2

k−2(1− γik−1) + O((1− γn

ik−1)2) + O((1− γn

ik−2)2),

≤ ck−2(1− γnik−2

) + O((1− γnik−1

)2),

following a Taylor expansion, where ck−2, c1k−2 and c2

k−2 are constants. Note that εk−2 > 0.

Proceeding in a similar fashion, one can construct p′1, ..., p′k−3 as well, wherein p′i − pn

i ≤ ci(1−γn

i1) + O((1 − γi1)

2), ci constant. Finally p′1 > p′2 > ... > p′k is ensured if εi < pni − pn

i−1,

i = 1, .., k − 1, which is guaranteed for n sufficiently large. Hence |pni − p′i| < c(1 − γn

i1),

i = 1, .., k, and this completes the proof of claim ii).

We will now prove the statement of the proposition. Denote the feasible set in equations (7),

(8)-(11) for the problem with risk-aversion parameter γn as T n, and for the risk-neutral case

as T . Following Proposition 1, pn ∈ T n and (p, r) ∈ T . Consider n > m so that γm <

γn. We will show that T n ⊂ T m. Consider any p ∈ T n. Then it satisfies constraint (11).

However, γm < γn implies that 1 ≥(

vi−pivi−pi+1

)γmi ≥

(vi−pi

vi−pi+1

)γni ≥ ri+1

riimplying that p ∈ T m.

This implies {pn}n≥m ⊂ T m. T m is a compact set, implying that there exists a subsequence

{nk}k∈N ⊂ N,m ≤ n1 < n2 < ... s.t. pnk → p̃, p̃ ∈ T m.

Suppose p̃ ∈ S and p̃ 6= p. Then ∀ε > 0, ∃M s.t. ∀k ≥ M , |pnki − p̃i| < ε. This implies

that the optimal revenue for problem Pnk , Rnk(pnk) ≤ R(p̃) + cε, where Rn(·) denotes the

revenue with risk-aversion parameter γn and R(·) denotes the revenue for the risk-neutral case

(under feasible price vectors). Now since the optimal solution to the risk-neutral problem P is

unique, R(p) > R(p̃) and for ε small enough, R(p) > Rnk(pnk). However, this would violate the

optimality of pnk , since for nk large enough, p is feasible for Snk . Then, from assumption 1, it

follows that p̃ ∈ S =⇒ p̃ = p.

Now suppose p̃ /∈ S. Then let δ = minp′∈S ||p̃ − p′||2 > 0. Note the for k large enough,

||pnk − p̃||2 < ε1, and ∃p′ ∈ S s.t. ||pnk − p′||2 < ε2. Now using the triangle inequality,

||pnk − p̃||2 ≥ ||p̃− p′||2−||pnk − p′||2 ≥ δ− ε2. Hence choosing ε1, ε2 to be such that ε1 + ε2 < δ,

we will achieve a contradiction. Hence p̃ ∈ S and consequently p̃ = p. In a similar fashion,

it is also possible to show that {pn} has a unique limit point. Moreover, this directly implies

that Rnk(pnk) → R(p) (Actually we know that Rnk(pnk) ≥ R(p) since p ∈ Snk , for k large, or

alternatively, by using Corollary 3 directly). ¤

Proof of Proposition 5 It is easy to verify that the following assignment,

ρi = 0, i = 1, 2, ..., N,

25

δil−1+1 = δil−1+2 = ... = δil , l = 1, 2, ..., k,

δj = δj+1 = ... = δN = 0,

δil =δil+1rl+1 + wil,il+1xil − wil,ilxil − εil

rl, l = 1, 2, ..., k,

is feasible for the LP (30)-(36). Constraints (31)-(32) and (34)-(36) are tight, while δik = − εikrk

implies that constraint (33) is also satisfied.

We will next show that (30)-(36) has a finite optimal solution by establishing that its dual

is itself feasible. The dual to (30)-(36) can be written as follows.

min Σiki=1µi[(wi,i+1 − wi,i)xi − εi] + ΣN−1

i=j µi(−εi) (41)

s.t. µi ≥ 0, i = 1, 2, ..., N − 1, ηi ≥ 0, i < ik, θi ≥ 0, i ≤ ik, (42)

λ ≥ 0, α1, ..., αi1 ≥ 0, βik ≥ 0, φj , ..., φN ≥ 0, νj , ...νN ≥ 0, (43)

π1λ− u1,1µ1 − η111/∈I + α1 = π1p1, (44)

πiλ− ui,iµi + ui−1,iµi−1 − ηi1i/∈I + ηi−11i−1/∈I + αi1i≤i1 = πipi, i = 2, 3, ..., ik − 1, (45)

πikλ− uik,ikµik + uik−1,ikµik−1 + ηik−11ik−1/∈I + αik1ik≤i1 = πikpik , (46)

πjλ− uj,jµj + uj−1,jµj−1 − φj = 0, (47)

πiλ− ui,iµj + ui−1,iµi−1 − φi = 0, i = j + 1, ..., N − 1, (48)

πNλ + uN−1,NµN−1 − φN = 0, (49)

µ1r1 − θ111/∈I = π1r1, (50)

− µi−1ri + µiri + θi−11i−1/∈I − θi1i/∈I = πiri, i = 2, ..., ik − 1, (51)

− µik−1rik + µikrik + θik−11ik−1/∈I − θik1ik /∈I + β = πikrik , (52)

θj−11j−1/∈I,j−1≤ik + νj = 0, (53)

νi = 0, i = j + 1, ..., N. (54)

Here λ is the dual variable associated with the capacity constraint (31), µi is the dual variable

associated with constraint (36), i = 1, ..., N−1, ηi is the dual variable associated with constraint

ρi ≥ ρi+1, i = 1, ..., N−1, θi is the dual variable associated with constraint δi ≥ δi+1, i = 1, ..., ik.

αi is the dual variable associated with the constraint ρi ≤ 0, i = 1, ..., i1, β is the dual variable

associated with the constraint δik ≤ 0, φi is the dual variable associated with the constraint

ρi ≥ 0, i = j, j + 1, ..., N , and νi is the dual variable associated with the constraint δi ≤ 0,

i = j, j + 1, ..., N .

Following Proposition 3, without loss of generality, we restrict attention to the case where

the optimal number of products to offer to risk-neutral customers k ≤ 2. By brute-force once

can verify that the following assignment of variables is feasible for (41)-(54), and therefore that

26

(30)-(36) has a finite feasible solution (details are omitted):

µi = Σil=1πl, i = 1, ..., N, (55)

θi = 0, i = 1, ..., N − 1, (56)

νi = 0, i = j, j + 1, ..., N, β = 0, (57)

η1 = 0, α1 = π1(v1 − λ), (58)

αi − ηi = −λπi + µivi − µi−1vi−1 − ηi−1, αi ≥ 0, ηi ≥ 0, αiηi = 0, 1 < i < i1, (59)

ηi1 = 0, αi1 = −λπi1 + µi1vi1 − µi1−1vi1−1 − ηi1−1, (60)

ηi = λ(Σil=i1+1πl) + µiivi1 − µivi, i1 < i < i2, (61)

φi = λπi + µi−1(vi−1 − vi), i = j, j + 1, ..., N, (62)

λ =

0, if k = 1,

(Σi2l=1πl)vi2

−(Σi1l=1πl)vi1

Σi2l=i1+1πl

, if k = 2.(63)

We next compute the optimal solution to (30)-(36). We consider the one-product and the two-

product case separately.

i) k = 1: In this case, wi,i+1−wi,i = 0, i = 1, ..., N−1. Also, εi = 0, i = 1, ..., N−1 would ensure

that the IC conditions are not violated. Together these imply that zero is feasible revenue for

the dual problem (41)-(54). However, this is also attained by setting δi = ρi = 0, i = 1, ..., N

in the primal problem. Hence by strong duality, this must be the optimal solution.

ii) k = 2: In this case, wi,i+1 − wi,i = 0, i = 1, ..., i1 − 1, i1 + 1, ..., N − 1. Also, we can set

εi = 0, i = 1, ..., i1 − 1, i1 + 1, ..., N − 1. This implies that a feasible dual revenue is given by

(Σi1l=1πl)[(wi1,i1+1 − wi1,i1)xi1 − εi1 ]. However, this is also attained by setting δ1 = δ2 = ... =

δi1 = (wi1,i1+1 − wi1,i1)xi1 − εi1 , δi = 0, i > i1, ρi = 0, i = 1, ..., N , in the primal solution.

Hence, by strong duality, this must be the optimal solution. ¤

Appendix B

Existence of optimal solution and constraint qualification: For completeness, we justify

the use of the Lagrangian approach. We observe that the objective (6) is continuous, and the

feasible sets (7), (8)-(12) and (7), (8)-(11) are compact. Hence, following Weierstrass theorem,

a maxima exists. Suppose the optimal solution to formulation (6), (8)-(12) involves offering

k ≤ N distinct products, where the products partition the customer types as in Lemma 4.

Define al = γil(vil − pl)γil−1rl, bl = γil(vil − pl+1)γil

−1rl+1, cl = (vil − pl)γil , dl = (vil − pl+1)γil ,

l = 1, ..., k − 1. Also define el = Σill=il−1+1πl, l = 1, ..., k. Then the matrix obtained by

differentiating the k − 1 downstream IC conditions and the capacity constraint is given as

27

follows.

−a1 b1 0 0 . . . 0 −d1 0 0 . . . 0 0

0 −a2 b2 0 . . . 0 c2 −d2 0 . . . 0 0...

. . ....

. . ....

0 0 0 0 . . . −ak−1 0 0 0 . . . ck−1 −dk−1

0 0 0 0 . . . 0 −e2 −e3 . . . −ek

The first k− 1 rows in this matrix correspond to the k− 1 downstream constraints, and the

kth corresponds to the capacity constraint. The first k− 1 columns correspond to the variables

p1, ..., pk−1, while the next k−1 columns correspond to variables r2, ..., rk. Note that this matrix

has rank k (since there are k linearly independent rows (the matrix is in row-echelon form, and

can easily be converted into reduced row-echelon form with k non-zero rows)), and hence the

constraint qualification condition is met for the Lagrangian in equation (38). In case we also

add the upstream constraints to the Lagrangian, as in equation (37), we observe that either

the upstream or the downstream constraint can be tight, but not both, and since the derivative

of any upstream constraint would lead to the same non-zero entries as the derivative of the

corresponding downstream constraint, the constraint qualification condition would be met.

Proof of algorithm 1: For risk-neutral customers, the required conditions are obtained from

the proof of Proposition 3. For risk-averse customers, the Lagrangian can be written as follows:

L = (Σi1l=1πl)p1 + (Σi2

l=i1+1πl)vi2r2 + µ ((vi1 − p1)γi1 − (vi1 − vi2)γi1 r2) + λ

(C − Σi1

l=1πl − (Σi2l=i1+1πl)r2

).

Differentiating with respect to p1 and r2 respectively, yields the following:

(Σi1l=1πl)− µγi1(vi1 − p1)γi1

−1 = 0,

(Σi2l=i1+1πl)vi2 − λ(Σi2

l=i1+1πl)− µ(vi1 − vi2)γi1 = 0.

There are two cases to consider: λ = 0 and λ > 0. λ = 0 implies that

p1 = vi1 −(

(Σi2l=i1+1πl)vi2γi1

(Σi1l=1πl)(vi1 − vi2)

γi1

) 11−γi1

r2 =

((Σi2

l=i1+1πl)vi2γi1

(Σi1l=1πl)(vi1 − vi2)

) γi11−γi1

.

The conditions p1 > vi2 and C ≥ (Σi1l=1πl) + (Σi2

l=i1+1πl)r2 require that

(Σi1l=1πl)(vi1 − vi2) > γi1(Σ

i2l=i1+1πl)vi2 , C ≥ (Σi1

l=1πl) + (Σi2l=i1+1πl)

((Σi2

l=i1+1πl)vi2γi1

(Σi1l=1πl)(vi1 − vi2)

) γi11−γi1

.

28

under which the optimal revenue is given by

R = (Σi1l=1πl)vi1 − (Σi1

l=1πl)

((Σi2

l=i1+1πl)vi2γi1

(Σi1l=1πl)(vi1 − vi2)

γi1

) 11−γi1

+ vi2(Σi2l=i1+1πl)

((Σi2

l=i1+1πl)vi2γi1

π1(vi1 − vi2)

) γi11−γi1

.

and which exceeds the one product revenue at price vi1 and vi2 under the sufficient condition

(Σi1l=1πl)(vi1 − vi2) > (Σi2

l=i1+1πl)vi2 . λ > 0 implies that

p1 = vi1 − (vi1 − vi2)

(C − (Σi1

l=1πl)

(Σi2l=i1+1πl)

) 1γi1

, r2 =C − (Σi1

l=1πl)

(Σi2l=i1+1πl)

.

The conditions p1 > vi2 and C ≤ (Σi1l=1πl) + (Σi2

l=i1+1πl)r2 require that

C < (Σi1l=1πl) + (Σi2

l=i1+1πl), C < (Σi1l=1πl) + (Σi2

l=i1+1πl)

((Σi2

l=i1+1πl)vi2γi1

(Σi1l=1πl)(vi1 − vi2)

) γi11−γi1

.

under which the optimal revenue is given by

(Σi1l=1πl)vi1 − (Σi1

l=1πl)(vi1 − vi2)

(C − (Σi1

l=1πl)

(Σi2l=i1+1πl)

) 1γi1

+ (Σi2l=i1+1πl)vi2

(C − (Σi1

l=1πl)

(Σi2l=i1+1πl)

),

and which exceeds the one product revenue at price vi1 and vi2 .

That the constraint qualification condition is met follows from 6. To see that the proposed

solution is indeed a maxima, write the Lagrangian as

L = (Σi1l=1πl)p1 + (Σi2

l=i1+1πl)vi2

(vi1 − p1

vi1 − vi2

)γi1

+ λ

(C − Σi1

l=1πl − (Σi2l=i1+1πl)

(vi1 − p1

vi1 − vi2

)γi1)

.

Differentiating with respect to p1, we obtain that

(Σi1l=1πl)−

(Σi2l=i1+1πl)vi2γi1(vi1 − p1)γi1

−1

(v1 − vi2)γi1−1 +

λ(Σi1l=1πl)γi1(vi1 − p1)γi1

−1

(v1 − vi2)γi1−1 = 0.

If λ = 0, then it is easy to verify that p1 is the same as obtained earlier and the second

derivative with respect to p1 is negative. If λ > 0, the tightness of the IC condition implies that

we obtain the same solution as before. Solving for λ and substituting to calculate the second

derivative with respect to p1, we find it to be negative, implying that the method does yield

revenue-maximizing solution.

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